Problem # p049 |

The rod of mass m in the figure to the left is in the process of sliding
along the wall and floor. Neglect friction.
The polar mass moment of inertia of the slender rod of length L and
mass m with respect to its center of mass is given by the equation : The purpose of this problem is to ultimately derive the differential equation governing the motion of the rod. You will have to address only the first five questions, see the special comment at the end of question 5. |

- Carefully sketch the Free Body Diagram of the rod. Label each arrow
representing a force and describe in a short sentence what the arrow
represents ( who is exerting it onto what ).
- Write out the equation for the x-component of the acceleration
of the center of mass ( at L/2 from either end ) in terms of
mass and x-components of the forces.
- Write out the equation for the y-component of the acceleration
of the center of mass ( at L/2 from either end )
in terms of mass and y-components of the forces.
- Write out the equation for the angular acceleration of the rod in
terms of the moment of the forces around the center of mass.
- Write out the vector equation for the absolute acceleration of
the rod at point B
in terms of the absolute acceleration of point A plus the relative
acceleration of B as seen by A. Do NOT assume that the angular velocity
of the rod is zero !!!

If you answer the above equations correctly you will get the full amount of credits for this problem. You may keep on going to earn some extra bonus credits.

- Write out the equation for the absolute acceleration of the center of mass
of the rod
in terms of the absolute acceleration of point A and the relative
acceleration of the center of mass as seen by A. Do NOT assume
that the angular velocity
of the rod is zero !!!
- By method of substitution ELIMINATE the forces, the 2 components of the
acceleration of the center of mass, and finally the acceleration
of point A from your collection of equations. You should be left
with a single equation relating the angular acceleration
d²Θ/dt²=α ,
the angular velocity dΘ/dt=ω, and the angle Θ to
each other. This equation describes
( as a differential equation ) the movement of the rod from some initial
condition.
Do not attempt to solve this equation; probably, it can be done only
on a computer.